Strong well-posedness for the phase-field Navier–Stokes equations in the maximal regularity class

In this paper we study the dynamics of vesicle membranes in incompressible viscous fluids. We prove existence and uniqueness of the local strong solution for this model coupling of the Navier–Stokes equations with a phase field equation in an $L_p - L_q$ setting. We transform the equation into a quasi-linear parabolic evolution equation and use the general theory proved by Prüss et al. Since the operator and the nonlinear term are analytic, we have that the solution is real analytic in time and space. At last it is shown that the variational strict stable solution is exponentially stable, provided the product of the viscosity coefficient and the mobility constant is large.

Keywords

phase field, Navier–Stokes, well-posedness, stability, vesicle membrane, fluid vesicle interaction, bending elastic energy

2010 Mathematics Subject Classification

35D35, 35Q30, 35Q35, 76D03, 76D05, 76T10

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The author would like to thank Professor Matthias Hieber for giving him the method of Section 2 in intensive lectures at the University of Tokyo. This work was supported by JSPS-DFG Japanese-German Graduate Externship. This work was done while the author is a research assistant of the university of Tokyo supported by JSPS through scientific grant Kiban S (26220702).

Received 15 April 2017

Accepted 22 November 2017

Published 29 March 2018